Problem with Rig. Part of one finger does not move with the mesh
If we press AltH in Edit mode we can see that you have a hidden bone, and if we check its Deform option we can see that it is activated, while I guess it should not because you probably don't want it to weigh on the fingers, so deactivate this option and it should work fine. Also, disable the Deform option for all the bones that are not supposed to weigh on the mesh, and maybe even reparent.